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Helix surfaces in Lorentzian Heisenberg group

Lorenzo Pellegrino

TL;DR

The paper develops a differential-geometric framework for helix (constant angle) surfaces in the Lorentzian Heisenberg group H3(τ) by deriving the Gauss-Codazzi equations tailored to H3(τ) and classifying minimal and constant mean curvature helix surfaces, providing explicit parametrizations. It extends the analysis to nonzero angle surfaces, giving complete descriptions for spacelike and timelike cases, and shows that minimal and CMC constant-angle surfaces must have zero angle unless special delta=1 cases apply. The results clarify the geometry of helices in Lorentzian homogeneous spaces and supply concrete parametrizations for planes and cylinders as well as new explicit models useful for geometric analysis in Lorentzian settings.

Abstract

In this work we investigate constant angle surfaces in the Lorentzian Heisenberg group $\htt$. After providing a complete description of the geometry of the ambient space, we perform the full classification of minimal and CMC helix surfaces in $\htt$, giving their explicit parametrizations. In addition, we investigate the constant angle spacelike and timelike surfaces for a Lorentzian metric on the Heisenberg group $H_3$ not treated before in the literature, first showing that such surfaces have constant Gaussian curvature and then obtaining their complete characterization.

Helix surfaces in Lorentzian Heisenberg group

TL;DR

The paper develops a differential-geometric framework for helix (constant angle) surfaces in the Lorentzian Heisenberg group H3(τ) by deriving the Gauss-Codazzi equations tailored to H3(τ) and classifying minimal and constant mean curvature helix surfaces, providing explicit parametrizations. It extends the analysis to nonzero angle surfaces, giving complete descriptions for spacelike and timelike cases, and shows that minimal and CMC constant-angle surfaces must have zero angle unless special delta=1 cases apply. The results clarify the geometry of helices in Lorentzian homogeneous spaces and supply concrete parametrizations for planes and cylinders as well as new explicit models useful for geometric analysis in Lorentzian settings.

Abstract

In this work we investigate constant angle surfaces in the Lorentzian Heisenberg group . After providing a complete description of the geometry of the ambient space, we perform the full classification of minimal and CMC helix surfaces in , giving their explicit parametrizations. In addition, we investigate the constant angle spacelike and timelike surfaces for a Lorentzian metric on the Heisenberg group not treated before in the literature, first showing that such surfaces have constant Gaussian curvature and then obtaining their complete characterization.

Paper Structure

This paper contains 9 sections, 12 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic geometry of the Lorentzian Heisenberg group
  4. Surfaces in Three-Dimensional Lorentzian Manifolds
  5. The structure equations for surfaces in ${H}_3(\tau)$
  6. Minimal helix surfaces in ${H}_3(\tau)$
  7. General helix surfaces in ${H}_3(\tau)$
  8. Spacelike helix surfaces in ${H}_3(\tau)$
  9. Timelike helix surfaces in ${H}_3(\tau)$

Key Result

Proposition 2.2

The Riemann curvature tensor ${R}^\tau$ of ${H}_3(\tau)$ is determined by for all vector fields $X,Y,Z$ on ${H}_3(\tau)$.

Theorems & Definitions (26)

  • Remark 2.1
  • Proposition 2.2
  • Lemma 2.3: CV1
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 4.1
  • ...and 16 more